login
A212850
Number of n X 3 arrays with rows being permutations of 0..2 and no column j greater than column j-1 in all rows.
13
1, 19, 163, 1135, 7291, 45199, 275563, 1666495, 10038331, 60348079, 362442763, 2175719455, 13057505371, 78354598159, 470156286763, 2821023814015, 16926401164411, 101559181827439, 609357415487563, 3656151466494175
OFFSET
1,2
COMMENTS
From Petros Hadjicostas, Aug 25 2019: (Start)
Both formulas below follow from the theory in the documentation of array A309951. We have Sum_{s = 0..A000041(3)} (-1)^s * A309951(3,s) * a(n-s) = 0, i.e., a(n) - 10*a(n-1) - 27*a(n-2) + 18*a(n-3) = 0 for n >= 4. This is a consequence of Eq. (6) on p. 248 of Abramson and Promislow (1978), where we let t=0 in the equation.
In the explicit formula by Vaclav Kotesovec below, a(n) = 6^n - 2*3^n + 1^n, the numbers 1, 3, 6 (that are raised to the n-th power) are the multinomial coefficients of the A000041(3) = 3 integer partitions of 3: 1 = 3!/3!, 3 = 3!/(1!2!), 6 = 3!/(1!1!1!).
(End)
LINKS
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6) on p. 248 (set t:=0).
FORMULA
Empirical: a(n) = 10*a(n-1) - 27*a(n-2) + 18*a(n-3).
Explicit formula: a(n) = 6^n - 2*3^n + 1. - Vaclav Kotesovec, May 31 2012
EXAMPLE
Some solutions for n=3:
1 2 0 2 1 0 0 2 1 1 2 0 1 2 0 2 1 0 1 2 0
2 0 1 2 0 1 2 0 1 2 0 1 0 2 1 2 0 1 1 0 2
0 2 1 0 1 2 2 1 0 2 1 0 2 0 1 0 2 1 0 2 1
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 28 2012
STATUS
approved